where the rate increases by a certain percentage each year. You can have R fit an exponential increase

by changing the link option from identity to log in the statement that invokes the Poisson regression:

glm(formula = Accidents ~ Year, family = poisson(link = “log”))

This produces the output shown in Figure 19-4 and graphed in Figure 19-5.

FIGURE 19-4: Output from an exponential trend Poisson regression.

Because of the log link used in this regression run, the coefficients are related to the logarithm of the

event rate. Thus, the relative rate of increase per year is obtained by taking the antilog of the

regression coefficient for Year. This is done by raising e (the mathematical constant 2.718…) to the

power of the regression coefficient for Year:

, which is about 1.11. So, according to an

exponential increase model, the annual accident rate increases by a factor of 1.11 each year —

meaning there is an 11 percent increase each year. The dashed-line curve in Figure 19-4 shows this

exponential trend, which appears to accommodate the steeper rate of increase seen after 2016.

Comparing alternative models

The bottom of Figure 19-4 shows the AIC value for the exponential trend model is 78.476, which is

about 3.2 units lower than for the linear trend model in Figure 19-2 (

). Smaller AIC values

indicate better fit, so the true trend is more likely to be exponential rather than linear. But you can’t

conclude that the model with the lower AIC is really better unless the AIC is about six units better. So

in this example, you can’t say for sure whether the trend is linear or exponential, or potentially another

distribution. But the exponential curve does seem to predict the high accident rates seen in 2020 and

2021 better than the linear trend model.